SET007 Axioms: SET007+34.ax
%------------------------------------------------------------------------------
% File : SET007+34 : TPTP v9.0.0. Released v3.4.0.
% Domain : Set Theory
% Axioms : Function Domains and Fraenkel Operator
% Version : [Urb08] axioms.
% English :
% Refs : [Mat90] Matuszewski (1990), Formalized Mathematics
% : [Urb07] Urban (2007), MPTP 0.2: Design, Implementation, and In
% : [Urb08] Urban (2006), Email to G. Sutcliffe
% Source : [Urb08]
% Names : fraenkel [Urb08]
% Status : Satisfiable
% Syntax : Number of formulae : 98 ( 17 unt; 0 def)
% Number of atoms : 419 ( 71 equ)
% Maximal formula atoms : 17 ( 4 avg)
% Number of connectives : 344 ( 23 ~; 2 |; 163 &)
% ( 50 <=>; 106 =>; 0 <=; 0 <~>)
% Maximal formula depth : 15 ( 6 avg)
% Maximal term depth : 4 ( 1 avg)
% Number of predicates : 52 ( 50 usr; 1 prp; 0-4 aty)
% Number of functors : 139 ( 139 usr; 103 con; 0-4 aty)
% Number of variables : 215 ( 147 !; 68 ?)
% SPC :
% Comments : The individual reference can be found in [Mat90] by looking for
% the name provided by [Urb08].
% : Translated by MPTP from the Mizar Mathematical Library 4.48.930.
% : These set theory axioms are used in encodings of problems in
% various domains, including ALG, CAT, GRP, LAT, SET, and TOP.
%------------------------------------------------------------------------------
fof(rc1_fraenkel,axiom,
? [A] :
( ~ v1_xboole_0(A)
& v1_fraenkel(A) ) ).
fof(cc1_fraenkel,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( m1_subset_1(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ) ).
fof(fc1_fraenkel,axiom,
! [A,B] : v1_fraenkel(k1_funct_2(A,B)) ).
fof(fc2_fraenkel,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> ( v1_finset_1(k1_funct_2(A,B))
& v1_fraenkel(k1_funct_2(A,B)) ) ) ).
fof(t1_fraenkel,axiom,
$true ).
fof(t2_fraenkel,axiom,
$true ).
fof(t3_fraenkel,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] :
( ~ v1_xboole_0(B)
=> ! [C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> ! [D] :
( ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) )
=> ! [E] :
( k7_relat_1(C,E) = k7_relat_1(D,E)
=> ! [F] :
( m1_subset_1(F,A)
=> ( r2_hidden(F,E)
=> k8_funct_2(A,B,C,F) = k8_funct_2(A,B,D,F) ) ) ) ) ) ) ) ).
fof(t4_fraenkel,axiom,
$true ).
fof(t5_fraenkel,axiom,
! [A,B] : r1_tarski(k1_funct_2(A,B),k1_zfmisc_1(k2_zfmisc_1(A,B))) ).
fof(t6_fraenkel,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D] :
( ( r1_tarski(C,B)
& r1_tarski(D,A) )
=> ( k1_funct_2(C,D) = k1_xboole_0
| ! [E] :
( m1_subset_1(E,k1_funct_2(C,D))
=> ( v1_funct_1(E)
& m2_relset_1(E,B,A) ) ) ) ) ) ).
fof(d1_fraenkel,axiom,
! [A] :
( v1_fraenkel(A)
<=> ! [B] :
( r2_hidden(B,A)
=> ( v1_relat_1(B)
& v1_funct_1(B) ) ) ) ).
fof(t7_fraenkel,axiom,
$true ).
fof(t8_fraenkel,axiom,
! [A] :
( ( v1_relat_1(A)
& v1_funct_1(A) )
=> v1_fraenkel(k1_tarski(A)) ) ).
fof(d2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(C)
& v1_fraenkel(C) )
=> ( m1_fraenkel(C,A,B)
<=> ! [D] :
( m1_subset_1(D,C)
=> ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) ) ) ) ) ).
fof(t9_fraenkel,axiom,
$true ).
fof(t10_fraenkel,axiom,
! [A,B,C] :
( ( v1_funct_1(C)
& v1_funct_2(C,A,B)
& m2_relset_1(C,A,B) )
=> m1_fraenkel(k2_setwiseo(k1_zfmisc_1(k2_zfmisc_1(A,B)),C),A,B) ) ).
fof(t11_fraenkel,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B] : m1_fraenkel(k1_funct_2(B,A),B,A) ) ).
fof(t12_fraenkel,axiom,
$true ).
fof(t13_fraenkel,axiom,
$true ).
fof(t14_fraenkel,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D] :
( ( r1_tarski(C,B)
& r1_tarski(D,A) )
=> ( k1_funct_2(C,D) = k1_xboole_0
| ! [E] :
( m1_subset_1(E,k1_funct_2(C,D))
=> ? [F] :
( m2_fraenkel(F,B,A,k1_fraenkel(B,A))
& k7_relat_1(F,C) = E ) ) ) ) ) ).
fof(t15_fraenkel,axiom,
! [A] :
( ~ v1_xboole_0(A)
=> ! [B,C,D] :
( m2_fraenkel(D,B,A,k1_fraenkel(B,A))
=> k7_relat_1(D,C) = k7_relat_1(D,k3_xboole_0(B,C)) ) ) ).
fof(t16_fraenkel,axiom,
! [A,B] :
( ( v1_finset_1(A)
& v1_finset_1(B) )
=> v1_finset_1(k1_funct_2(A,B)) ) ).
fof(t17_fraenkel,axiom,
! [A] :
( v1_fraenkel(A)
=> ! [B] :
( r1_tarski(B,A)
=> v1_fraenkel(B) ) ) ).
fof(s23_fraenkel,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s23_fraenkel)
=> p1_s23_fraenkel(A,A) )
& ! [A] :
( m1_subset_1(A,f1_s23_fraenkel)
=> ! [B] :
( m1_subset_1(B,f1_s23_fraenkel)
=> ! [C] :
( m1_subset_1(C,f1_s23_fraenkel)
=> ( ( p1_s23_fraenkel(A,B)
& p1_s23_fraenkel(B,C) )
=> p1_s23_fraenkel(A,C) ) ) ) ) )
=> ! [A] :
( m1_subset_1(A,f1_s23_fraenkel)
=> ~ ( r2_hidden(A,f2_s23_fraenkel)
& ! [B] :
( m1_subset_1(B,f1_s23_fraenkel)
=> ~ ( r2_hidden(B,f2_s23_fraenkel)
& p1_s23_fraenkel(B,A)
& ! [C] :
( m1_subset_1(C,f1_s23_fraenkel)
=> ( ( r2_hidden(C,f2_s23_fraenkel)
& p1_s23_fraenkel(C,B) )
=> p1_s23_fraenkel(B,C) ) ) ) ) ) ) ) ).
fof(s24_fraenkel,axiom,
? [A] :
( m1_subset_1(A,k5_finsub_1(f1_s24_fraenkel))
& ! [B] :
( m1_subset_1(B,f1_s24_fraenkel)
=> ( r2_hidden(B,A)
<=> ? [C] :
( m1_subset_1(C,f2_s24_fraenkel)
& r2_hidden(C,f3_s24_fraenkel)
& B = f4_s24_fraenkel(C)
& p1_s24_fraenkel(B,C) ) ) ) ) ).
fof(s26_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s26_fraenkel)
=> ~ ( r2_hidden(A,f3_s26_fraenkel)
& ! [B] :
( m1_subset_1(B,f2_s26_fraenkel)
=> ~ p1_s26_fraenkel(A,B) ) ) )
=> ? [A] :
( v1_funct_1(A)
& v1_funct_2(A,f1_s26_fraenkel,f2_s26_fraenkel)
& m2_relset_1(A,f1_s26_fraenkel,f2_s26_fraenkel)
& ! [B] :
( m1_subset_1(B,f1_s26_fraenkel)
=> ( r2_hidden(B,f3_s26_fraenkel)
=> p1_s26_fraenkel(B,k8_funct_2(f1_s26_fraenkel,f2_s26_fraenkel,A,B)) ) ) ) ) ).
fof(s27_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s27_fraenkel)
=> ~ ( r2_hidden(A,f3_s27_fraenkel)
& ! [B] :
( m1_subset_1(B,f2_s27_fraenkel)
=> ~ p1_s27_fraenkel(A,B) ) ) )
=> ? [A] :
( m2_fraenkel(A,f1_s27_fraenkel,f2_s27_fraenkel,k1_fraenkel(f1_s27_fraenkel,f2_s27_fraenkel))
& ! [B] :
( m1_subset_1(B,f1_s27_fraenkel)
=> ( r2_hidden(B,f3_s27_fraenkel)
=> p1_s27_fraenkel(B,k8_funct_2(f1_s27_fraenkel,f2_s27_fraenkel,A,B)) ) ) ) ) ).
fof(dt_m1_fraenkel,axiom,
! [A,B,C] :
( m1_fraenkel(C,A,B)
=> ( ~ v1_xboole_0(C)
& v1_fraenkel(C) ) ) ).
fof(existence_m1_fraenkel,axiom,
! [A,B] :
? [C] : m1_fraenkel(C,A,B) ).
fof(dt_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ! [D] :
( m2_fraenkel(D,A,B,C)
=> ( v1_funct_1(D)
& v1_funct_2(D,A,B)
& m2_relset_1(D,A,B) ) ) ) ).
fof(existence_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ? [D] : m2_fraenkel(D,A,B,C) ) ).
fof(redefinition_m2_fraenkel,axiom,
! [A,B,C] :
( ( ~ v1_xboole_0(B)
& m1_fraenkel(C,A,B) )
=> ! [D] :
( m2_fraenkel(D,A,B,C)
<=> m1_subset_1(D,C) ) ) ).
fof(dt_k1_fraenkel,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> m1_fraenkel(k1_fraenkel(A,B),A,B) ) ).
fof(redefinition_k1_fraenkel,axiom,
! [A,B] :
( ~ v1_xboole_0(B)
=> k1_fraenkel(A,B) = k1_funct_2(A,B) ) ).
fof(s1_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s1_fraenkel)
=> ( p1_s1_fraenkel(A)
=> p2_s1_fraenkel(A) ) )
=> r1_tarski(a_0_0_fraenkel,a_0_1_fraenkel) ) ).
fof(s2_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s2_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s2_fraenkel)
=> ( p1_s2_fraenkel(A,B)
=> p2_s2_fraenkel(A,B) ) ) )
=> r1_tarski(a_0_2_fraenkel,a_0_3_fraenkel) ) ).
fof(s3_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s3_fraenkel)
=> ( p1_s3_fraenkel(A)
<=> p2_s3_fraenkel(A) ) )
=> a_0_4_fraenkel = a_0_5_fraenkel ) ).
fof(s4_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s4_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s4_fraenkel)
=> ( p1_s4_fraenkel(A,B)
<=> p2_s4_fraenkel(A,B) ) ) )
=> a_0_6_fraenkel = a_0_7_fraenkel ) ).
fof(s5_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s5_fraenkel)
=> f2_s5_fraenkel(A) = f3_s5_fraenkel(A) )
=> a_0_8_fraenkel = a_0_9_fraenkel ) ).
fof(s6_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s6_fraenkel)
=> ( p1_s6_fraenkel(A)
=> f2_s6_fraenkel(A) = f3_s6_fraenkel(A) ) )
=> a_0_10_fraenkel = a_0_11_fraenkel ) ).
fof(s7_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s7_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s7_fraenkel)
=> f3_s7_fraenkel(A,B) = f4_s7_fraenkel(A,B) ) )
=> a_0_12_fraenkel = a_0_13_fraenkel ) ).
fof(s8_fraenkel,axiom,
( ( ! [A] :
( m1_subset_1(A,f1_s8_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s8_fraenkel)
=> ( p1_s8_fraenkel(A,B)
<=> p2_s8_fraenkel(A,B) ) ) )
& ! [A] :
( m1_subset_1(A,f1_s8_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s8_fraenkel)
=> f3_s8_fraenkel(A,B) = f3_s8_fraenkel(B,A) ) ) )
=> a_0_14_fraenkel = a_0_15_fraenkel ) ).
fof(s9_fraenkel,axiom,
( ( k7_relat_1(f4_s9_fraenkel,f3_s9_fraenkel) = k7_relat_1(f5_s9_fraenkel,f3_s9_fraenkel)
& ! [A] :
( m1_subset_1(A,f1_s9_fraenkel)
=> ( r2_hidden(A,f3_s9_fraenkel)
=> ( p1_s9_fraenkel(A)
<=> p2_s9_fraenkel(A) ) ) ) )
=> a_0_16_fraenkel = a_0_17_fraenkel ) ).
fof(s10_fraenkel,axiom,
r1_tarski(a_0_18_fraenkel,f1_s10_fraenkel) ).
fof(s11_fraenkel,axiom,
( ! [A] :
( r2_hidden(A,a_0_19_fraenkel)
=> p2_s11_fraenkel(A) )
=> ! [A] :
( m1_subset_1(A,f1_s11_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s11_fraenkel)
=> ( p1_s11_fraenkel(A,B)
=> p2_s11_fraenkel(f3_s11_fraenkel(A,B)) ) ) ) ) ).
fof(s12_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s12_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s12_fraenkel)
=> ( p1_s12_fraenkel(A,B)
=> p2_s12_fraenkel(f3_s12_fraenkel(A,B)) ) ) )
=> ! [A] :
( r2_hidden(A,a_0_20_fraenkel)
=> p2_s12_fraenkel(A) ) ) ).
fof(s13_fraenkel,axiom,
a_0_21_fraenkel = a_0_23_fraenkel ).
fof(s14_fraenkel,axiom,
a_0_24_fraenkel = a_0_26_fraenkel ).
fof(s15_fraenkel,axiom,
a_0_27_fraenkel = a_0_29_fraenkel ).
fof(s16_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s16_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s16_fraenkel)
=> ~ ( p1_s16_fraenkel(A,B)
& ! [C] :
( m1_subset_1(C,f1_s16_fraenkel)
=> ~ ( p2_s16_fraenkel(C,B)
& f3_s16_fraenkel(A,B) = f3_s16_fraenkel(C,B) ) ) ) ) )
=> r1_tarski(a_0_30_fraenkel,a_0_31_fraenkel) ) ).
fof(s17_fraenkel,axiom,
r1_tarski(a_0_32_fraenkel,f2_s17_fraenkel) ).
fof(s18_fraenkel,axiom,
r1_xboole_0(a_0_33_fraenkel,f2_s18_fraenkel) ).
fof(s19_fraenkel,axiom,
( ! [A] :
( m1_subset_1(A,f1_s19_fraenkel)
=> ! [B] :
( m1_subset_1(B,f2_s19_fraenkel)
=> ( p2_s19_fraenkel(A,B)
<=> ( B = f4_s19_fraenkel
& p1_s19_fraenkel(A,B) ) ) ) )
=> a_0_34_fraenkel = a_0_35_fraenkel ) ).
fof(s20_fraenkel,axiom,
a_0_36_fraenkel = a_0_37_fraenkel ).
fof(s21_fraenkel,axiom,
( v1_finset_1(f2_s21_fraenkel)
=> v1_finset_1(a_0_38_fraenkel) ) ).
fof(s22_fraenkel,axiom,
( ( v1_finset_1(f3_s22_fraenkel)
& v1_finset_1(f4_s22_fraenkel) )
=> v1_finset_1(a_0_39_fraenkel) ) ).
fof(s25_fraenkel,axiom,
( ( v1_finset_1(f3_s25_fraenkel)
& v1_finset_1(f4_s25_fraenkel)
& ! [A] :
( m2_fraenkel(A,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
=> ! [B] :
( m2_fraenkel(B,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
=> ( k7_relat_1(A,f3_s25_fraenkel) = k7_relat_1(B,f3_s25_fraenkel)
=> f5_s25_fraenkel(A) = f5_s25_fraenkel(B) ) ) ) )
=> v1_finset_1(a_0_40_fraenkel) ) ).
fof(fraenkel_a_0_0_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_0_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s1_fraenkel)
& A = f2_s1_fraenkel(B)
& p1_s1_fraenkel(B) ) ) ).
fof(fraenkel_a_0_1_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_1_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s1_fraenkel)
& A = f2_s1_fraenkel(B)
& p2_s1_fraenkel(B) ) ) ).
fof(fraenkel_a_0_2_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_2_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s2_fraenkel)
& m1_subset_1(C,f2_s2_fraenkel)
& A = f3_s2_fraenkel(B,C)
& p1_s2_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_3_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_3_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s2_fraenkel)
& m1_subset_1(C,f2_s2_fraenkel)
& A = f3_s2_fraenkel(B,C)
& p2_s2_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_4_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_4_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s3_fraenkel)
& A = f2_s3_fraenkel(B)
& p1_s3_fraenkel(B) ) ) ).
fof(fraenkel_a_0_5_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_5_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s3_fraenkel)
& A = f2_s3_fraenkel(B)
& p2_s3_fraenkel(B) ) ) ).
fof(fraenkel_a_0_6_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_6_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s4_fraenkel)
& m1_subset_1(C,f2_s4_fraenkel)
& A = f3_s4_fraenkel(B,C)
& p1_s4_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_7_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_7_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s4_fraenkel)
& m1_subset_1(C,f2_s4_fraenkel)
& A = f3_s4_fraenkel(B,C)
& p2_s4_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_8_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_8_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s5_fraenkel)
& A = f2_s5_fraenkel(B)
& p1_s5_fraenkel(B) ) ) ).
fof(fraenkel_a_0_9_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_9_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s5_fraenkel)
& A = f3_s5_fraenkel(B)
& p1_s5_fraenkel(B) ) ) ).
fof(fraenkel_a_0_10_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_10_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s6_fraenkel)
& A = f2_s6_fraenkel(B)
& p1_s6_fraenkel(B) ) ) ).
fof(fraenkel_a_0_11_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_11_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s6_fraenkel)
& A = f3_s6_fraenkel(B)
& p1_s6_fraenkel(B) ) ) ).
fof(fraenkel_a_0_12_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_12_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s7_fraenkel)
& m1_subset_1(C,f2_s7_fraenkel)
& A = f3_s7_fraenkel(B,C)
& p1_s7_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_13_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_13_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s7_fraenkel)
& m1_subset_1(C,f2_s7_fraenkel)
& A = f4_s7_fraenkel(B,C)
& p1_s7_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_14_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_14_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s8_fraenkel)
& m1_subset_1(C,f2_s8_fraenkel)
& A = f3_s8_fraenkel(B,C)
& p1_s8_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_15_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_15_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s8_fraenkel)
& m1_subset_1(C,f2_s8_fraenkel)
& A = f3_s8_fraenkel(C,B)
& p2_s8_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_16_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_16_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s9_fraenkel)
& A = k8_funct_2(f1_s9_fraenkel,f2_s9_fraenkel,f4_s9_fraenkel,B)
& p1_s9_fraenkel(B)
& r2_hidden(B,f3_s9_fraenkel) ) ) ).
fof(fraenkel_a_0_17_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_17_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s9_fraenkel)
& A = k8_funct_2(f1_s9_fraenkel,f2_s9_fraenkel,f5_s9_fraenkel,B)
& p2_s9_fraenkel(B)
& r2_hidden(B,f3_s9_fraenkel) ) ) ).
fof(fraenkel_a_0_18_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_18_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s10_fraenkel)
& A = B
& p1_s10_fraenkel(B) ) ) ).
fof(fraenkel_a_0_19_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_19_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s11_fraenkel)
& m1_subset_1(C,f2_s11_fraenkel)
& A = f3_s11_fraenkel(B,C)
& p1_s11_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_20_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_20_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s12_fraenkel)
& m1_subset_1(C,f2_s12_fraenkel)
& A = f3_s12_fraenkel(B,C)
& p1_s12_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_21_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_21_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f3_s13_fraenkel)
& A = B
& r2_hidden(B,a_0_22_fraenkel)
& p2_s13_fraenkel(B) ) ) ).
fof(fraenkel_a_0_22_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_22_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s13_fraenkel)
& m1_subset_1(C,f2_s13_fraenkel)
& A = f4_s13_fraenkel(B,C)
& p1_s13_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_23_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_23_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s13_fraenkel)
& m1_subset_1(C,f2_s13_fraenkel)
& A = f4_s13_fraenkel(B,C)
& p1_s13_fraenkel(B,C)
& p2_s13_fraenkel(f4_s13_fraenkel(B,C)) ) ) ).
fof(fraenkel_a_0_24_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_24_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s14_fraenkel)
& A = f2_s14_fraenkel(B)
& r2_hidden(B,a_0_25_fraenkel)
& p1_s14_fraenkel(B) ) ) ).
fof(fraenkel_a_0_25_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_25_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s14_fraenkel)
& A = B
& p2_s14_fraenkel(B) ) ) ).
fof(fraenkel_a_0_26_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_26_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s14_fraenkel)
& A = f2_s14_fraenkel(B)
& p2_s14_fraenkel(B)
& p1_s14_fraenkel(B) ) ) ).
fof(fraenkel_a_0_27_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_27_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s15_fraenkel)
& m1_subset_1(C,f2_s15_fraenkel)
& A = f3_s15_fraenkel(B,C)
& r2_hidden(B,a_0_28_fraenkel)
& p1_s15_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_28_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_28_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s15_fraenkel)
& A = B
& p2_s15_fraenkel(B) ) ) ).
fof(fraenkel_a_0_29_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_29_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s15_fraenkel)
& m1_subset_1(C,f2_s15_fraenkel)
& A = f3_s15_fraenkel(B,C)
& p2_s15_fraenkel(B)
& p1_s15_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_30_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_30_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s16_fraenkel)
& m1_subset_1(C,f2_s16_fraenkel)
& A = f3_s16_fraenkel(B,C)
& p1_s16_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_31_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_31_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s16_fraenkel)
& m1_subset_1(C,f2_s16_fraenkel)
& A = f3_s16_fraenkel(B,C)
& p2_s16_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_32_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_32_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s17_fraenkel)
& A = f3_s17_fraenkel(B)
& r2_hidden(f3_s17_fraenkel(B),f2_s17_fraenkel)
& p1_s17_fraenkel(B) ) ) ).
fof(fraenkel_a_0_33_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_33_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s18_fraenkel)
& A = f3_s18_fraenkel(B)
& p1_s18_fraenkel(B)
& ~ r2_hidden(f3_s18_fraenkel(B),f2_s18_fraenkel) ) ) ).
fof(fraenkel_a_0_34_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_34_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s19_fraenkel)
& m1_subset_1(C,f2_s19_fraenkel)
& A = f3_s19_fraenkel(B,C)
& p2_s19_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_35_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_35_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s19_fraenkel)
& A = f3_s19_fraenkel(B,f4_s19_fraenkel)
& p1_s19_fraenkel(B,f4_s19_fraenkel) ) ) ).
fof(fraenkel_a_0_36_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_36_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s20_fraenkel)
& m1_subset_1(C,f2_s20_fraenkel)
& A = f3_s20_fraenkel(B,C)
& C = f4_s20_fraenkel
& p1_s20_fraenkel(B,C) ) ) ).
fof(fraenkel_a_0_37_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_37_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s20_fraenkel)
& A = f3_s20_fraenkel(B,f4_s20_fraenkel)
& p1_s20_fraenkel(B,f4_s20_fraenkel) ) ) ).
fof(fraenkel_a_0_38_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_38_fraenkel)
<=> ? [B] :
( m1_subset_1(B,f1_s21_fraenkel)
& A = f3_s21_fraenkel(B)
& r2_hidden(B,f2_s21_fraenkel) ) ) ).
fof(fraenkel_a_0_39_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_39_fraenkel)
<=> ? [B,C] :
( m1_subset_1(B,f1_s22_fraenkel)
& m1_subset_1(C,f2_s22_fraenkel)
& A = f5_s22_fraenkel(B,C)
& r2_hidden(B,f3_s22_fraenkel)
& r2_hidden(C,f4_s22_fraenkel) ) ) ).
fof(fraenkel_a_0_40_fraenkel,axiom,
! [A] :
( r2_hidden(A,a_0_40_fraenkel)
<=> ? [B] :
( m2_fraenkel(B,f1_s25_fraenkel,f2_s25_fraenkel,k1_fraenkel(f1_s25_fraenkel,f2_s25_fraenkel))
& A = f5_s25_fraenkel(B)
& r1_tarski(k2_funct_2(f1_s25_fraenkel,f2_s25_fraenkel,B,f3_s25_fraenkel),f4_s25_fraenkel) ) ) ).
%------------------------------------------------------------------------------